Context.Nearly all members of the Vesta family cross the orbits of (4) Vesta, one of the most massive asteroids in the main belt, and
some of them approach it closely. When mutual velocities
during such close encounters are low, the trajectory of
the small body can be gravitationally deflected, consequently
changing its heliocentric orbital elements.
While the effect of a single close encounter may be small, repeated
close encounters may significantly change the proper element
distribution of members of asteroid families.

Aims.We develop a model of the long-term effect of close encounters
with massive asteroids, so as to be able to predict how far former
members of the Vesta family could have drifted away from the family.

Methods.We first developed a new symplectic integrator that simulates both
the effects of close encounters and the Yarkovsky effect.
We analyzed the results of a simulation involving a fictitious
Vesta family, and propagated the asteroid proper element distribution
using the probability
density function (pdf hereafter), i.e. the function that
describes the probability of having an encounter that
modifies a proper element x by Δx,
for all the possible values of Δx. Given any
asteroids' proper element distribution at time t, the
distribution at time may be predicted if the pdf is
known (Bachelier 1900, Théorie de la spéculation; Hughes 1995,
Random Walks and Random Environments, Vol. I).

Results.We applied our new method to the problem of V-type asteroids outside
the Vesta family (i.e., the 31 currently known asteroids
in the inner asteroid belt that have
the same spectral type of members as the Vesta family, but that are outside
the limits of the dynamical family) and
determined that at least
ten objects have a significant diffusion probability over the minimum
estimated age of the Vesta family of 1.2 Gyr (Carruba et al. 2005, A&A, 441, 819).
These objects can therefore be explained
in the framework of diffusion via repeated close encounters with (4) Vesta
of asteroids originally closer to the parent body.

Conclusions.We computed diffusion probabilities at the location
of four of these asteroids for various initial conditions,
parametrized by values of initial ejection velocity Vej. Based
on our results, we believe the Vesta family age is (1200 ± 700) Myr old,
with an initial ejection velocity of (240 ± 60) m/s.

Key words: minor planets, asteroids / celestial mechanics

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